A note on weighted consecutive Davenport constant

Let $G$ be a group and $A\subseteq [1,\exp (G)-1]$. We define the constant ${\mathsf{C}}_A(G),$ which is the least positive integer $\mathrm{\ell }$ such that every sequence over $G$ of length at least $\mathrm{\ell }$ has an $A$-weighted consecutive product-one subsequence. In this paper, among other things, we prove that ${\mathsf{C}}_A(C_n^2)=4$ with $A=[1,n-1],$ and $\mathsf{C}(H\times K)=\mathrm{\mid }H\mathrm{\mid }\mathrm{\mid }K\mathrm{\mid }$, where $H$ is a finite abelian group and $K$ is a metacyclic group.